TITLE: Linearly independent functions
QUESTION [5 upvotes]: Show that the set consisting of the functions
$$x, e^x, e^{-x}$$
on $\mathbb R$ is linearly independent.
So I have the equation $$ax + be^x + ce^{-x} = 0$$ and I want to show that this is only satisfied when $a = b = c = 0$
Letting x = 0, $b + c = 0$
Letting x = 1, $a + be + ce^{-1} = 0$
Letting x = -1, $-a + be^{-1} + ce = 0$
Using these equations as columns of a matrix I have
$\begin{bmatrix}
0 & 1 & 1 \\
1 & e & e^{-1} \\
-1 & e^{-1} & e
\end{bmatrix}$
$\begin{bmatrix}
1 & e & e^{-1} \\
0 & 1 & 1 \\
-1 & e^{-1} & e
\end{bmatrix}$
$\begin{bmatrix}
1 & e & e^{-1} \\
0 & 1 & 1 \\
0 & e^{-1}+e & e^{-1}+e
\end{bmatrix}$
But if I now subtract (e^{-1}+e) times row 2 from row 3 I will get all zeros in the last row meaning linear dependence. So what have I done wrong?

REPLY [7 votes]: This is most naturally attacked using the Wronskian, 
$$W(f_1,\ldots,f_n)(x) = \det\left(
\begin{array}{ccc}
f_1(x)  & \cdots & f_n(x) \\
f_1'(x) & \cdots & f'_n(x) \\
\vdots  & \vdots    & \vdots \\
f_1^{(n-1)}(x)   & \cdots & f_n^{(n-1)}(x)
\end{array}\right),$$
where $f_k^{(m)}(x)$ is the $m$th derivative of $f_k(x)$.
If the Wronskian does not vanish, the functions $f_k(x)$ are linearly independent. 
We find 
$$W(x,e^x,e^{-x}) = \det\left(
\begin{array}{ccc}
x   & e^x   & e^{-x} \\
1   & e^x   & -e^{-x} \\
0   & e^x   & e^{-x} 
\end{array}\right) = 2x.$$
Therefore, the functions are linearly independent on $\mathbb{C}\setminus\{0\}$. (And therefore on $\mathbb{R}\setminus\{0\}$.)

REPLY [7 votes]: Several people have posted rather general methods for attacking this problem.  Here is an alternative approach that is rather specific to this case.  Suppose that $a$, $b$, and $c$ are numbers and that
$$
ax + b e^x + c e^{-x} = 0, \qquad x \in \mathbb{R}.
$$
Divide both sides of this by $e^x$ and take the limit as $x \to \infty$ to deduce $b = 0$.  Once you know that, dividing both sides of the above by $x$ and taking the limit as $x \to \infty$, you deduce $a = 0$.  And $c=0$ clearly follows from there.
The same basic idea can be used slightly more generally, to prove linear independence of functions of different "growth rates" without solving systems of equations.